Resolving the Isoparametric Concept with Compatibility

Comment: This post was inspired by conversations I have had during the past few days at a workshop I recently attended. I apologize ahead of time for its problem specific nature and overall lack of generality. Nonetheless, it is simply my attempt at resolving some inconsistencies and misconceptions in the field of computational mechanics.

I had the pleasure of attending a workshop in Pavia, Italy this past week entitled "Non-Standard Discretizations for Partial Differential Equations." One of the main focuses of this workshop was the subject of compatible discretizations. Compatible discretizations are ones which preserve certain properties of the underlying partial differential equation such as topology, conservation, symmetries, and positivity structures and maximum principles. The main tools behind the construction of these methods as often quite mathematical in nature (e.g., exact sequences), and they have not quite taken hold yet in the engineering community. Nonetheless, these methods have very nice properties and typically are much more accurate than standard numerical techniques.

In the engineering community, the utilization of the isoparametric concept is pretty much a religious practice. The isoparametric concept simply states that one uses the same discretization for the geometry as well as for the analysis. This is an especially useful property for problems in nonlinear structural mechanics as it allows one to represent exactly all rigid body modes (translations and rotations). However, most compatible discretizations do not preserve the isoparametric property, and many mathematicians argue they should not. Some in fact state the adherence to the isoparametric concept is just a consequence of inertia and misunderstanding. In certain circumstances, I would have to agree with these mathematicians. However, in others, I would have to disagree. In fact, in some important instances, I would contend that the isoparametric concept is one of compatability.

Consider elasticity. Elasticity has many meanings, and we need to respect that. In particular, linear elasticity and nonlinear elasticity are completely different animals. Linear elasticity is a theory of fields. It is a theory of vectors in Cartesian space. In a sense, it is a theory of Eulerian flow. Nonlinear Lagrangian elasticity, on the other hand, is a theory of shape representation. One is not solving for vector fields on top of a given geometry. Instead, in nonlinear elasticity, one is solving for the geometry itself. So many discretizations for nonlinear elasticity are based on simple extensions of those for linear elasticity, but from the compatibility standpoint this makes absolutely zero sense! These two problems have very different geometrical, topological, and symmetry properties.

I would agree that the isoparametric concept has little meaning in the discretization of fields defined on top of a given geometry, as in linear elasticity. After all, in linear elasticity, one is not really changing the surface or volume. However, it has clear meaning in nonlinear elasticity, where one involves a moving geometry. The isoparametric concept preserves the most important geometric invariant of all: conservation of shape under translation and rotation. This ensures there is no energy associated with rigid body modes, an incredibly important conservation law for nonlinear elasticity. This property ensures no spurious modes just as compatible edge elements ensure no spurious modes for electromagnetics. In this sense, the isoparametric concept is akin to compatibility. It is a key component of compatibility of shape.

All in all, I would contend that we should know what we are dealing with before we decide what is compatible or not. If we are dealing with fields, we should worry about compatibility of fields. If we are dealing instead with shape, we should worry instead about compatibility of shape and consequently the isoparametric concept.